The Stacks project

Remark 90.7.10. For a category $\mathcal{C}$ we denote by $\text{CofSet}(\mathcal{C})$ the category of cofibered sets over $\mathcal{C}$. It is a $1$-category isomorphic the category of functors $\mathcal{C} \to \textit{Sets}$. See Remarks 90.5.2 (11). The completion and restriction functors restrict to functors $\widehat{~ } : \text{CofSet}(\mathcal{C}_\Lambda ) \to \text{CofSet}(\widehat{\mathcal{C}}_\Lambda )$ and $|_{\mathcal{C}_\Lambda } : \text{CofSet}(\widehat{\mathcal{C}}_\Lambda ) \to \text{CofSet}(\mathcal{C}_\Lambda )$ which we denote by the same symbols. As functors on the categories of cofibered sets, completion and restriction are adjoints in the usual 1-categorical sense: the same construction as in Remark 90.7.9 defines a functorial bijection

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }(G|_{\mathcal{C}_\Lambda }, F) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }(G, \widehat{F}) \]

for $F \in \mathop{\mathrm{Ob}}\nolimits (\text{CofSet}(\mathcal{C}_\Lambda ))$ and $G \in \mathop{\mathrm{Ob}}\nolimits (\text{CofSet}(\widehat{\mathcal{C}}_\Lambda ))$. Again the map $\widehat{F}|_{\mathcal{C}_\Lambda } \to F$ is an isomorphism.

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